Es sei ${\displaystyle {}{\mathfrak {n}}}$ ein maximales Ideal in einem kommutativen Ring ${\displaystyle {}R}$. Es sei ${\displaystyle {}R_{\mathfrak {n}}}$ die Lokalisierung von ${\displaystyle {}R}$ an ${\displaystyle {}{\mathfrak {n}}}$ und es sei ${\displaystyle {}{\mathfrak {m}}={\mathfrak {n}}R_{\mathfrak {n}}}$ das maximale Ideal von ${\displaystyle {}R_{\mathfrak {n}}}$. Zeige

${\displaystyle {}R/{\mathfrak {n}}=R_{\mathfrak {n}}/{\mathfrak {m}}}$